Beverley Rahman

2022-02-18

After having read abstract concepts of algebraic curves, I have trouble dealing with actual examples. For instance, why is the $\varphi =\frac{y}{x}$ a rational function on the curve $F={y}^{2}+y+{x}^{2}$? I know that any rational function on this curve should be of the form $\left\{\varphi =\frac{f}{g}:f,g\in \frac{K\left[x,y\right]}{\left(F\right)},g\ne 0\right\}$, but what do I need to actually check to show that this is a rational function on F? Any help will be good

copausc20

Your definition of a rational function is just fine for where you're at and your function exactly fits it. To see this, note that you need to verify that f=y and g=x are elements of $\frac{K\left[x,y\right]}{\left(F\right)}$ and that $g\ne 0$ (as a function). But this is clear.